Editing Reflexive space (section)

{{Short description|Locally convex topological vector space}}
In the area of mathematics known as [[functional analysis]], a '''reflexive space''' is a [[locally convex]] [[topological vector space]] for which the canonical evaluation map from <math>X</math> into its [[bidual]] (which is the [[strong dual]] of the strong dual of <math>X</math>) is a [[homeomorphism]] (or equivalently, a [[TVS isomorphism]]). 
A [[normed space]] is reflexive if and only if this canonical evaluation map is [[surjective]], in which case this (always linear)  evaluation map is an [[isometric isomorphism]] and the normed space is a [[Banach space]]. Those spaces for which the canonical evaluation map is surjective are called [[semi-reflexive]] spaces. 

In 1951, [[Robert C. James|R. C. James]] discovered a Banach space, now known as [[James' space]], that is {{em|not}} reflexive (meaning that the canonical evaluation map is not an isomorphism) but is nevertheless isometrically isomorphic to its bidual (any such [[isometric isomorphism]] is necessarily {{em|not}} the canonical evaluation map). So importantly, for a Banach space to be reflexive, it is not enough for it to be isometrically isomorphic to its bidual; it is the canonical evaluation map in particular that has to be a homeomorphism.

Reflexive spaces play an important role in the general theory of [[locally convex]] TVSs and in the theory of [[Banach space]]s in particular.  [[Hilbert space]]s are prominent examples of reflexive Banach spaces.  Reflexive Banach spaces are often characterized by their geometric properties.